Cumulative tutorial 09
Fitting Rayleigh damping, and checking it
Anchor the damping at two frequencies, then find out what every other mode received.
The three-storey building from the previous tutorial is to be run by direct integration, which needs a damping MATRIX rather than modal ratios. Rayleigh damping is to be fitted to give 5% at the first and third modes. Fit it — then check what the second mode actually got.
What you are given
- ω₁ = 19.903 rad/s, ω₂ = 55.75 rad/s, ω₃ = 80.585 rad/s
- Target damping 5% at modes 1 and 3
- C = a₀M + a₁K
Tutorial 09
Work the problem
4 questions
Question 1
What is the mass-proportional coefficient a₀?
Question 2
What is the stiffness-proportional coefficient a₁?
Question 3
What damping ratio does the second mode actually receive, as a percentage?
Question 4
By what percentage is the second mode under-damped relative to the 5% target?
What this establishes
Rayleigh damping is exact at two frequencies and wrong everywhere else, and 'wrong' has a direction that depends on where the mode sits. Fit it, then tabulate the ratio every retained mode actually received. That table takes a minute and is the difference between a damping model and a damping assumption.